Existence and uniqueness of solutions of Hahn difference equations

Hamza, Alaa E.; Ahmed, Samah M

doi:10.1186/1687-1847-2013-316

Cited by 38 publications

(22 citation statements)

References 15 publications

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“…The exponential, trigonometric, and hyperbolic functions associated with D β were presented in [9]. The existence and uniqueness of solutions of the first-order β-initial value problem were established in [11]. In [6], the existence and uniqueness of solutions of the β-Cauchy problem of the second-order β-difference equations were proved.…”

Section: Introductionmentioning

confidence: 99%

Theory of nth-order linear general quantum difference equations

2018

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In this paper, we derive the solutions of homogeneous and non-homogeneous nth-order linear general quantum difference equations based on the general quantum difference operator D β which is defined by D β f (t) = (f (β(t))-f (t))/(β(t)-t), β(t) = t, where β is a strictly increasing continuous function defined on an interval I ⊆ R that has only one fixed point s 0 ∈ I. We also give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of these equations. Furthermore, we present the fundamental set of solutions when the coefficients are constants, the β-Wronskian associated with D β , and Liouville's formula for the β-difference equations. Finally, we introduce the undetermined coefficients, the variation of parameters, and the annihilator methods for the non-homogeneous β-difference equations.

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Section: Introductionmentioning

confidence: 99%

Theory of nth-order linear general quantum difference equations

2018

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“…In 2013, Malinowska and Martins [19] presented the generalized transversality conditions for the Hahn quantum variational calculus. Next, Hamza and Ahmed [20,21] established the theory of linear Hahn difference equations. The authors also study the existence and uniqueness of solution for the initial value problems for Hahn difference equations by employing the method of successive approximations; in addition, they proved Gronwall's and Bernoulli's inequalities with respect to the Hahn difference operator and investigated the mean value theorems for this calculus.…”

Section: Introductionmentioning

confidence: 99%

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

Patanarapeelert

Sitthiwirattham

2017

Discrete Dynamics in Nature and Society

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The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixedpoint theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

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“…In the literature there exist some papers studying Hahn difference equations. In [16], the authors studied the theory of linear Hahn difference equations. They also studied the existence and uniqueness of the solutions of initial value problems defined via Hahn difference equations.…”

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confidence: 99%